Newtonian limit FRW

The newtonian limit for general relativity exists only for the asymptotically flat spacetimes.
FRW case definitely not asymptotically flat. SO the newtonian limit should not exist for it.
However we have Newtonian theory of cosmology in homogeneous and isotropic universe.
So my question is is it possible to obtain Newtonian cosmology from taking appropriately defined limit of FRW cosmology???

Secondly if we consider for instance Oppenheiner-Sneider collapse.
Collapse of the star which remains spatially homogeneous.
In this case The metric inside star is given by FRW. Outside it`s given by schwarzschild metric.
Hence this situation is asymptotically flat. So newtonian limit would make sence in this case.

If we try to take limit inside star
metric is given by
(ds)**2 = -(dt)**2+ space part

we need to compare that with
(ds)**2 = -(1+2*Phi)(dt)**2+ space part

where Phi is newtonian gravitational potential

compare these two
it implies that

PHi=0
hence essentially no gravity inside the star.

This is clearly not right..
Since there would be gravity inside star in Newtonian limit.

The inside of the (collapsing ) matter is a different space-time from the exterior. The interior is not asymptotically flat but the exterior is.

If the matter is not collapsing the interior may be modelled by the interior Shwarzschild space-time, in which a Newtonian approximation can be made.

One should also bear in mind that the source of the gravity in FLRW is non-interacting dust. So it is not surprising that no attraction can be found between the motes. If the source was interacting matter, the result could be very different.

The inside of the (collapsing ) matter is a different space-time from the exterior. The interior is not asymptotically flat but the exterior is.

Can you please elaborate this.

The notion which I have in mind about asymptotically flat spacetime is that as r--> infinity
the metric should tend to be minkoskian.
When we take r--> infinity limit we are into region of spacetime which is described by Schwartschild and it tends to be minkoski. Hence spacetime as a whole is Asymptotically flat.
Small region of it is described by FRW should not make any difference.

The idea of 'asymptotic flatness' has no meaning in FLRW co-ordinates. The scalar curvature of the Schwarzschild space-time is zero every where, but in FLRW it is non-zero and the same everywhere BUT it depends on time.

Newtonian approximations are just that - they are approximations and cannot be expected to hold everywhere in GR. The idea of a FRLW universe is completely outside Newtons theory which could not describe such a thing.

In this case we don`t have FRW universe.
Ofcourse if entire universe were described by FRW metric then Newtonian limit doesnt make sense.
But here we are concerned with spacetime ...only part of it is described by FRW and in my opinion it happens to be Asymptotically flat.

But as you have pointed out probably we should not use FRW coordinates.

Since in FRW coordinates we are essentially dealing with freely falling observers And hence in newtonian limit there would be no gravity. This is same as sitting in freely falling elevator near the surface of the earth and then commenting that ball in the elevator does not fall under gravity.Hence there is no gravity.

Certainly if you use the Newtonian concept of gravitational field strength, then there's apparently no such thing in FLRW. There's no time dilation that depends on spatial coordinates, which is different from Schwarzschild. So you could say that 'gravity' is the same everywhere in FLRW.

But I dispute that the patched FLRW and Schwarzschild space-time can be thought of as one homogenous thing, but it's hardly an important issue is it ?

In this case basically you start with spherically symmetric metric in synchronous coordinate system. And then solve Einstein`s equation both inside and outside star. And it just turns out that solution of the Einstein`s equation inside the star is FRW and outside it`s Scwarschild.
So there is nothing special about pasting FRW with Scharzschild.

To start with you have spherically symmetric spacetime that is asymptotically free.

If this problem is worked out in Schwarschild like coordinates in stead of FRW-like then maybe it would be possible to take newtonian limit inside as well as outside the star.

Yeah right.
Time dilation has to be same everywhere since FRW is spatially homogeneous and isotropic at any given epoch.
And Time dilation not being same would violate homogeneity.

Regarding your calculation of the time dilation
When we write down the FRW metric in usual form we work in synchronous comoving system.
This coordinate system is chosen so that three velocity of comoving dust is always zero.
Hence last three terms in that expression are zero.
Hence proper time is same as time coordinate . Hence there is no time dilation atall in this coordinate system.

Secondly section on Newtonian coordinate system in MTW says that while taking Newotonian limit you must work in so called "Newtonian Co-ordinates".
These are coordinates are ones in which metric can be decomposed into flat minkoski plus small perturbation.
This can`t be done in ususal coordinates .Hence we must look for appropriate coordinate system in which that can be done.

I am not sure how one should go about searching for such a coordinate system.